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The partial differential equation $c^{2} u_{x x}-u_{t t}=0$ where $c$ is a constant is called the wave equation. Show that $u(x, t)$ satisfies the wave equation for the given value of $c$.$$c=3, u(x, t)=5(x+3 t)^{3}$$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 2

Partial Derivatives

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12:15

In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

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The partial differential e…

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The wave equation $c^{2} \…

way we want to show this is a solution to the wave equation. We first, we're gonna have to find the partials with respect, accent T or the second partial for Tax and T. So let's go ahead and do that. So first, start with partial respect. X. So when we're taking that derivative, we would need to use channel. So first power. So five times three, we get 15 x plus three t squared del Beidle X of X plus three teams and I'll three t we're assuming is a constant with respect to X, so that would be zero. And then the derivative of X is just one. So that would give us 15 X plus three great t squared. And then we can go ahead and take the partial this again to get our second partial, and so would use chain rule again. So 1st 15 times 2. 30 x plus three t and then we take the drugs on the inside, which already found to be one. So that would be just, uh, 30. And I'll go ahead and distribute So 30 x waas 90 80. Yeah. Now we can come over here, do the same thing, but with respect to you, so you t will be so we use general again. So it would be 15 x plus three t square. But now we have Dele Bide lt of X Plus three Tian inside And then this. Here, um we're assuming X is a constant. So that would be zero. And then that would be three. So we're just gonna multiply that by three out here 45 x plus three t squared and then we can go ahead and take the second partial retrospective team. And so chain rule. So 90 times X plus three t take the road on inside, which should also be third again and then Slough give us to 70. Um, if we go ahead and distribute that that would give to 70 The 2 70 times three is 1 80 thio ex ante. Now we could go ahead and put this into that equation we had before, so it's supposed to be C squared. You x x minus you TT So our C is three. So that would be nine times early. X plus 90 t and then minus to 70 x plus 1 18. And if we distribute the nine that would give us to 70 x waas 1 18 and we'll know this that to seventies Cancel out the one eighties cancel out, and so then that would give us zero. So this does indeed satisfied the wave equation.

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